Mathematical Physics

Title:The principle of stationary nonconservative action for classical mechanics and field theories

Abstract: We further develop a recently introduced variational principle of stationary
action for problems in nonconservative classical mechanics and extend it to
classical field theories. The variational calculus used is consistent with an
initial value formulation of physical problems and allows for time-irreversible
processes, such as dissipation, to be included at the level of the action. In
this formalism, the equations of motion are generated by extremizing a
nonconservative action $\mathcal{S}$, which is a functional of a doubled set of
degrees of freedom. The corresponding nonconservative Lagrangian contains a
potential $K$ which generates nonconservative forces and interactions. Such a
nonconservative potential can arise in several ways, including from an open
system interacting with inaccessible degrees of freedom or from integrating out
or coarse-graining a subset of variables in closed systems. We generalize
Noether's theorem to show how Noether currents are modified and no longer
conserved when $K$ is non-vanishing. Consequently, the nonconservative aspects
of a physical system are derived solely from $K$. We show how to use the
formalism with examples of nonconservative actions for discrete systems
including forced damped harmonic oscillators, radiation reaction on an
accelerated charge, and RLC circuits. We present examples for nonconservative
classical field theories. Our approach naturally allows for irreversible
thermodynamic processes to be included in an unconstrained variational
principle. We present the nonconservative action for a Navier-Stokes fluid
including the effects of viscous dissipation and heat diffusion, as well as an
action that generates the Maxwell model for viscoelastic materials, which can
be easily generalized to more realistic rheological models. We show that the
nonconservative action can be derived as the classical limit of a more complete
quantum theory.